As mentioned in the introduction,
there is a conjecture (not restricted to triangular numbers)
due to Heide, Saxl, Tiep and Zalesski [1]},
which says that
for any \(n\ne 2,4,9\)  there is some character \(\psi \in {\rm Irr}(S_n)\) 
such that \(\psi ^2\)  contains all \(\chi \in {\rm Irr}(S_n)\)  as constituents.
The Saxl conjecture suggests a special candidate for triangular numbers \(n\) .
Apart from computational results for (relatively) small \(k\) ,
there are already many contributions towards this conjecture that
confirm that several families of constituents occur in the square
(see e.g. [2]}, [3]}, [4]}, [5]}, [6]}).
